Conservation of mass for a steady axisymmetric flow field in the cylindrical $(r, z)$ coordinates is:
$$\frac{1}{r} \frac{\partial\left(r V_{r}\right)}{\partial r}+\frac{\partial V_{z}}{\partial z}=0 $$
Here, $V_{r}$ and $V_{z}$ are radial and axial components of velocity, respectively.
Which one of the following options is correct if $\psi$ is the stream function?
- $V_{r}=\frac{\partial \psi}{\partial z}$ and $V_{z}=-\frac{1}{r} \frac{\partial \psi}{\partial r}$
- $V_{r}=\frac{1}{r} \frac{\partial \psi}{\partial z}$ and $V_{z}=-\frac{1}{r} \frac{\partial \psi}{\partial r}$
- $V_{r}=\frac{1}{r} \frac{\partial \psi}{\partial z}$ and $V_{z}=\frac{1}{r} \frac{\partial \psi}{\partial r}$
- $V_{r}=\frac{1}{r} \frac{\partial \psi}{\partial z}$ and $V_{z}=-\frac{\partial \psi}{\partial r}$
